Counting: There's more to it than 1-2-3!

This is a continuation of yesterday’s article, Quantification, which is just a fancy word for figuring out how much you’ve got. To recap, there are three developmental stages of quantification:

• Global quantification – child relies on perception.
• One-to-one correspondence – child matches objects one for one.
• Counting.

Check yesterday’s article, Quantification: How children learn to count, for a review of the first two stages.

Today we’ll focus on the third stage: Counting.

Counting is a not a trivial thing for little ones to grasp. There are three stages a child must come to understand.

1. Knowing the right order of the words. At this stage, it’s about the same as reciting a poem. It’s not the ABC poem but the 123 poem. This is called stable order counting.
   
   Have you ever observed a child count the fingers on one hand and end up with more than 5? They counted properly: 1-2-3-4-5-6, but they counted one finger twice. These children understand stable order counting – all the words were said in the right order - but they haven’t moved on to the next stage, which is…

   
   

2. Knowing that each object may be counted only one time, with one number word said for each object counted. This is an example of one-to-one correspondence: one word for each object.
   
   
3. The third stage is knowing that the last number represents the total. This is formally known as cardinality. This requires the understanding that the last number has two roles: the name assigned to the last item, and the name for the entire set. The pinkie finger is number 5, and the total number of fingers is 5.

To demonstrate the difference between one-to-one correspondence and cardinality, here’s a favorite story:

There was a little boy who was eating cookies with his mom. He counted them in the plate: 1, 2, 3. After he ate one his mom asked, “How many are there now?” “Three!” he replied. Mom asked how could that be, if he ate one? He said that he ate “1”, but not “2” or “3”.

This boy understood that there’s a word for each cookie, but he didn’t understand cardinality. The words for each cookie change when the quantity changes.

Fortunately for him, his mother accepted his answer and didn’t pursue a correct answer. These mistakes should be viewed as windows into a child’s knowledge rather than opportunities for correcting.

Once children grasp these 3 concepts:

• correct word order,
• one word for one object,
• the last word represents the total

then they start using counting as a method of quantification.

And so you see, counting is NOT as easy as 1-2-3!

To your child's success,

Rita

Quantification: How children learn to count

“It’s as easy as 1-2-3!” How often have we heard that phrase? Is counting really that easy? Let’s take a look behind the scenes at how a child learns to count. There’s more to it than you might think.

Today I’ll write about the developmental stages a child goes through which lead to counting. Tomorrow I’ll write about the developmental stages of counting itself.

Quantification is the formal name for the concept of figuring out how many things you’ve got. Counting is just one method of quantification.

There are 3 developmental stages a child goes through when learning to quantify.

1. Global Quantification. This is the stage where children are influenced by perceptions. They make a visual approximation of the quantity they’re trying to match.

• If a child wants as many goldfish as another, she’ll take a handful that seems to match the quantity.
• If she’s asked to take as many blocks, she’ll line them up side-by-side until her line is about the same length, without regard to the actual number of blocks.

2. One-to-one Correspondence. Children still use visual or tactile perceptions as in the previous stage, but in a more logical manner.

• Now the child will line the blocks up one-for-one, matching one new block for each of the original ones.

3. Counting. Children use counting as a method to quantify as they become comfortable with the first two stages, since counting relies on global quantification and 1-to-1 correspondence skills. Counting itself has several stages of understanding. Stay tuned for details in tomorrow’s Matinee Muse. :)

These stages will overlap, depending on the number of objects. A child who can count to 5 might use counting when there are only a few objects, one-to-one correspondence when there are more than a few, and global quantification when there are many objects.

Now what?
Now that you know the details behind your child’s ever-spinning gears, what do you do with this information? Give them appropriate activities! Here are a few ideas to spark your imagination.

Global Quantification ideas.

• Talk in terms of “more” and “less”, “many” and “few”, “big”, “little”, “tall”, “small”, “bigger” and so on. “Same” is a useful word, too.
• When your child helps with snack, toys, tea party or bathtub play, ask to have More or Less than them.
• Or, ask that every teddy be given the same amount.

Counting might not occur yet at this stage. If you're curious, you could ask, “How do you know (it’s the same)?” and observe how he answers. This will give you insight into the developmental stage he’s in. It’s an opportunity to listen, not necessarily to correct.

One-to-one correspondence ideas.

• Ask that every seat at the table get one plate, fork, cup, etc.
• Assemble a group of dinosaurs and give them each a piece of Lego dinner.
• Line up a row of blocks and put a shell on top of each.
• Gather 8 similar toys and give your child 8 pretend "coins". Trade a toy for each coin.
• Give your child 6 dolls and 6 hats. She will naturally pair them up.
• Put small rubber fruit counters on each square of a checkerboard.
• Place a goldfish on each square of a checkerboard placemat while waiting to be served.
• Place an object in each hole of an egg carton. The objects should fill the hole; otherwise your child might be tempted to put several in each hole.

You get the idea! Now it’s time to have fun with whatever you have handy that excites your child.

To your child's success,
Rita

Little Kids & Big Numbers

What’s the big deal with being able to count to 100 in preschool?

The beginning of February is when many schools celebrate their 100th Day of School. Since some of my blog subscribers have preschoolers, this blog entry is about the importance of big numbers for little kids.

I’ll say it as plainly as I can: I think big numbers are over-rated for toddlers. Unless a child has a rough concept of how many 100 is, it’s pretty useless to be able to count to 100. I consider it the equivalent of being able to recite a poem, the 1-2-3 poem. Until a child is able to grasp how much 1, or 2 or 58 is, big numbers just aren’t that big a deal. Counting to 10 or 20 is plenty.

What comes after 10?
Once a child can count to 10, the next step is to play around with the little numbers. What’s more, 1 or 2? Hmmm, if I were 2, that would be a terribly boring question!

How about: “I have 3 apple slices. You can have some, and I’ll have some. Would you like the bowl with 1 or the bowl with 2?” Of course this is not one of those math word problems to be done in their head; this is a real-life problem to be acted out at snack time.

This makes math part of a child’s daily life, and it uses the smallest of numbers. There’s really no need to jump ahead – even to 6 or 8 slices – until a child has a firm grasp of smaller numbers.

Having a solid grounding of how to manipulate small numbers will prepare a child for greater mathematical success than being able to recite to 100.

So the next time your friend boasts about how high their toddler can count, give them a smile and say “That’s nice!” but resist the pressure to compete. Big numbers are over-rated.

To your child's success,
Rita

What IS math, anyhow?

One of my favorite parts of conducting my Family Math Nights is chatting with parents. (I also love playing games with the kids, but that’s a different story.) Here are two very different conversations I had at two recent Family Math Nights.

Conversation 1:
Remember the old TV commercial where a grouchy granny opens a hamburger bun and grumbles, “Where’s the beef?”

I hosted a Family Math Night a few weeks ago where a grandfather sought me out to have a word with me. He began, “I want you to know I’m a true math skeptic. I see a lot of children and parents having fun here, but I don’t see any math. What you have here is logic, reasoning and spatial games, but I don’t see any math.”

“Ah! You’re looking for number games,” I replied.

Yes, that was it. We had a lively chat on what math is, how it can come in forms other than numbers, and how it can go beyond “number math”. I then pointed him to the stations which had a bit of “number math”, and checking in with him later he was a very happy camper.

Conversation 2:
At last night’s Family Math Night, I met a father who was very interested in discussing whether one of the games could be described in a computer algorithm. He’s a Computer Scientist, and my background is in engineering, so this was a fascinating conversation.

I then told him of the grandfather’s skepticism. He said that in India, where he grew up, that would be call arithmetic, and that arithmetic is only a small part of mathematics.

If math is more than numbers, what exactly is it?
There are 10 different math standards which students in grades Kindergarten through high school must be taught each year. These are set by NCTM, the National Council of Teachers of Mathematics, the folks who define the math standards for the U.S.

The first standard is “Number and Operations”, which could just as well be called “Arithmetic”. If this is just one of 10 standards, what is all the rest of Mathematics about? Here is NCTM’s complete list:

• Number and Operations
• Algebra
• Geometry
• Measurement
• Data Analysis and Probability
• Problem Solving
• Reasoning and Proof
• Communication
• Connections
• Representation

That’s a lot of math to cover every year!

A few of my personal favorites for younger students are what's known as Logic and Reasoning and Spatial Skills. These would fall under NCTM’s “Reasoning and Proof”, “Problem Solving” and “Geometry”. Pre-algebra skills are a lot of fun at this age, too.

I’ll write more about the standards in my upcoming blogs. In the meantime, what is math to you?

To your child's success,
Rita

More 100th Day of School Fun - NIM

One of my all-time favorite math games to play with children, NIM, can be adapted to a 100th Day of School game.

100th Day of School NIM
A game for 2 players, to be played in your head or with a calculator if you must.

Starting at zero, players take turns adding any number they choose from 1 to 9 to the ongoing total. The player to reach exactly 100 wins. For example:
Player 1: 0+9=9
Player 2: 9+1=10
Player 1: 10+5=15
Player 2: 15+5=20
Many more turns and calculations later:
Player 1: 90+8=98
Player 2: 98+2=100 And Player 2 wins!

NIM is considered a “deterministic game”. This means that this is a pure strategy game, and that if both players know the winning strategy, the winner can be determined before the game begins.

What this means is that you must:
1) Always let the child choose whether to go first or second.
2) Always play to win. (Otherwise it’s a random game with little hope for learning the winning strategy.)

An Important Note to Parents:
If you figure out the winning strategy, for heaven’s sake don’t tell your child!! This is stealing the pleasure of figuring it out for themselves right out from under them. You no doubt will feel a sense of accomplishment if you uncover the strategy on your own; let your child experience that wonderful feeling, too.

If this 100 game is too difficult for your child, here’s the standard NIM game which is within reach of the youngest elementary school students. It may take them years to figure out the winning strategy (that’s okay, really!) but the calculations will be doable.

Basic NIM
Start with 12 objects.
Players take turns removing 1, 2 or 3 objects at a time, announcing how many objects are left at the end of the turn.
Whoever removes the last object(s), wins.

This is a great travel game with pennies. Play it with the sugar bags the next time you’re waiting for your food at a restaurant. But remember, don’t give away the answer or you’ll have to think up a new game the next time you go out to eat!

Once your child has figured out how to win with 12, start with 13 or more. Can he or she still beat the game?

If you really must see your child master the strategy, I’ll be happy to share my tips on how to unfold the game’s secrets. As anyone who has been to one of my Family Math Nights knows, I don’t give away answers -- only tips on how to think about the solution.

Enjoy!
Rita